You can claim any dependents who legally qualify to be dependents. The rules are here. https://www.irs.gov/uac/about-publication-501
You don't get tax deductions for care expenses. You may be eligible for a credit if you pay for dependent care (day care) so that you and your spouse if married can go to work.
The costs you pay for a child rarely are involved in determining whether that child is your dependent.
There are two kinds of dependent, qualifying child and qualifying relative.
A qualifying child dependent is someone who is
1. your child, stepchild, grandchild, or a child of your brother or sister (niece or nephew), and
2, is under age 19, or under age 24 and also a full time student, and
3. lives with you more than half the year, and
4. does not earn more than half their own support.
How much you pay for their support doesn't really matter as long as they don't provide more than half on their own. (This is rare, of course, think child actors etc.)
If a child is not a qualifying child dependent they might be a qualifying relative dependent. That means,
1. not a qualifying child dependent of any other taxpayer, and
2. you provide more than half their support, and
3. does not have more than $4050 of taxable income of their own, and
4. lives with you the entire year, or is your biological child, grandchild, or parent.
Only in this case is it necessary to add up the dependent's total support costs and figure out if you pay more or less than half.
Certain complicating factors arise if there is shared custody between unmarried parents who live together, or with parents (never married, divorced or separated) who live apart.
Turbotax has a very good interview that will help you in this regard.
